Direct Energy, 2018a

294 13.2 Preliminary Ideas
Assuming V doesn't depend on θ or φ, the integral is separable.
(ˆ π
θ=0
ˆ 2π
φ=0
) ˆ 1
ˆ 1
sin θdθdφ V (r)r 2 dr =4π V (r)r 2 dr (13.16)
r=0
r=0
A sphere ofradius r has volume 4 3 πr3 .
13.2.2 Notation
Writing this text without overloading variables has been a challenge. For
example, V is the logical choice for denoting voltage, volume, and velocity.
Up until now, the context oered clues to the meaning ofsymbols. However
in this chapter, we will encounter equations involving both energy and
electric eld, equations involving both voltage and volume, and equations
involving both mass and momentum. To help avoid confusion from the
notation, Table 13.1 shows an excerpt ofthe variable list from Appendix
A. This table does not list all quantities we will encounter. However, it
highlights some ofthe more confusing ones.
In this chapter, we will encounter many quantities which vary with position.
We will not encounter any quantities which vary with time. Therefore,
voltage is denoted by a capital letter, not a lowercase letter. Voltage
is a function of r, which denotes position in spherical coordinates. Assume
that the origin ofthe coordinate system is at the center ofthe atom
under consideration. Voltage is always specied with respect to some reference
level called ground, so assume this zero volt reference level occurs
at r = ∞. Also assume there is no θ or φ dependence ofthe voltage.
Therefore, V ( −→ r )=V (r) represents voltage.